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# l13 - the state equation in the time domain we can take the...

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16.06 Lecture 13 More State Space Modeling and Transfer Function Matrices John Deyst October 2, 2003 Today’s Topics 1. More state space modeling. 2. Laplace Transforms for vector/matrix differential equations. 3. An example 1

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A special case for state space models is when there are as many zeros as poles. For example if we have Recall that we first break G ( s ) into two parts The first transfer function relates the input r to the intermediate output x 1 . In the time domain this implies the D.E. 2
So the state D.E.’s are Hence Implying the following block diagrams 3

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However now the output is which, in the time domain is If we go back to the block diagram we see that we must feed the ˙ x 2 signal forword to get the output We can then write the equation for w 4
Laplace Transforms of State Vectors With appropraite definitions of terms we can usefully apply Laplace Transform methods to vector/matrix differential equations. Thus we define the Laplace Transform of a state vector as It then follows naturally that the L.T. of the derivative of x ( t ) is 5

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Thus if we have the

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Unformatted text preview: the state equation in the time domain we can take the L.T. of both sides to obtain which can also be written as where I is the identity matrix 6 We can then solve for the L.T. of the state as Now we also have the output equation which can also be transformed to obtain the L.T. of the output vector Upon substitution of the equation for the L.T. of the state obtains 7 If we are only interested in the response to inputs we can obtain a matrix transfer function where Example Earlier we turned the scalar transfer function into a state space model and found the following matrices 8 Then Its inverse is Note that the denominater is the left hand side of the characteristic equation Also 9...
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